The future of cancer treatment optimization will depend on the seamless integration of experimental research and theoretical modeling. A significant challenge in oncology is the limited understanding of how therapeutic interventions influence the tumor microenvironment and how immune responses contribute to tumor regression. This knowledge gap is particularly pronounced in aggressive cancers such as triple-negative breast cancer (TNBC), which lacks standard biomarkers and remains difficult to treat. Our collaborative research focuses on developing a quantitative framework to investigate coupled immune and chemotherapeutic interventions. Specifically, we employ novel radiolabeled T-cell tracking to gain detailed insights into immunomodulation over time and use these data to inform optimal therapy scheduling strategies. In this talk, we will present ordinary differential equation (ODE) models that describe the dynamics of combined therapy, comparing scenarios with and without T-cell observations. Additionally, we will discuss the broader implications of this modeling approach, including its potential extension to organ-level analyses and the application of optimal control techniques to refine treatment strategies. Through this integrated approach, we aim to improve the predictive power of cancer treatment models and contribute to more effective, personalized therapeutic protocols.

Interfacial instabilities in volatile liquid thin films on a hydrophobic substrate can lead to complex droplet dynamics, such as droplet merging, splitting, and transport. Controlling these fundamental droplet behaviors is essential for digital microfluidics in biomedical and electric applications. In this talk, I will present our recent work on mean-field control of thin film droplets. We design an optimal control problem by formulating droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. As an example, we consider a thin volatile liquid film laden with an active suspension, where control is achieved through its activity field. Numerical examples, including droplet transport, bead-up/spreading, and merging/splitting, demonstrate the effectiveness of the proposed control mechanism.  

Over the summer, I attended the Early Graduate Research in Applied Mathematics Conference at Caltech in Pasadena, CA. This week-long event brought together graduate students from across the country to connect, share research interests, and build community. Through professional development sessions and hands-on activities, participants explored a wide range of applied mathematics topics, with special focus on using neural ordinary differential equations (ODEs) to model biological processes. In this talk, I will detail my experience investigating these topics and highlight the many professional development opportunities offered at this event.

I will then discuss my current research, which involves the development of an ordinary differential equation model to examine the dynamic relationship between inflammation, pain, and sleep in patients with sickle cell disease (SCD). The model consists of four compartments including pro-inflammation, anti-inflammation, pain, and sleep, with feedback mechanisms that capture clinically observed interactions. Specifically, pain disrupts sleep, poor sleep promotes inflammation, and elevated inflammation amplifies pain. The overall goal of this work is to identify interventions that could mitigate the impact of key feedback loops, which would provide insight into potential strategies for reducing pain crises in SCD.

Coral reefs are a vital part of marine ecosystems across the planet, providing shelter and food to sea creatures and protecting coastlines from storms and floods. However, reefs are threatened by coral bleaching, which can be caused by a number of factors, including long-term heat stress. Bleaching results from symbiotic algae leaving the coral, and may eventually result in host death. While coral can recover from some of these bleaching events, this process may take months or years. As these events occur more frequently, it is essential that we study coral growth and heat-induced bleaching dynamics. To do so, we construct a dynamic compartmental model consisting of four nonlinear ordinary differential equations to track changes in coral host, symbiont, nutrient, and nutrient reserve populations. We simulate growth and decay outcomes as well as the time to bleaching, death, or recovery for a range of temperature and exposure time scenarios. We will calibrate the model to experimental and environmental data to determine the behavior of different corals under heat stress and the feasibility of predicting future bleaching events.

The circadian clock shapes nearly 24-hour periodic rhythms throughout the body, from the activity of individual cells to our daily sleep/wake cycles. These rhythms can be self-sustained (for example, people still show circadian patterns even in total darkness), but they are also strongly influenced by the environment, especially light. Mathematical models of circadian rhythms can provide insight into many intriguing phenomena, including jet lag, mid-afternoon fatigue, and how animals sense the changing seasons. In this talk, I will present recent work using mathematical models to investigate the daily regulation of dopamine, a neurotransmitter involved in motivation, attention, and mood. Our models reveal how circadian control of dopamine metabolism leads to time-of-day effects of dopamine reuptake inhibitors commonly used to treat fatigue, ADHD, and depression. Furthermore, by modeling the collective activity of dopaminergic neurons, we identify intrinsic ~4-hour ultradian oscillations that arise independently of circadian input. These limit cycle oscillations exhibit flexible periods that can be modulated by pharmacological intervention. Overall, this talk will highlight the utility of mechanistic models for generating testable hypotheses about neuromodulatory systems.

The spongy moth (Lymantria dispar dispar) is an invasive forest pest that has caused significant ecological damage across the United States. Its invasion front is shaped by a number of factors, particularly temperature in both the northern and southern regions. With ongoing climate change, areas that were previously uninhabitable may become increasingly favorable for moth population establishment and expansion, while other areas may experience thermal stress limiting persistence. This study develops a temperature-driven population model to analyze how temperature affects the spongy moth population dynamics along the range margin. This model incorporates temperature effects on biological concepts such as fecundity, stage-structured survival rates, and the Allee threshold. By integrating these temperature-driven parameters into a discrete population framework, the model provides a foundation to explore how warming conditions may alter persistence and expansion along the invasion front.

 Mathematical modeling depends on observations of real-world dynamics, assumptions about system structure, and refinement through comparison with data. Sparse data-driven discovery offers a way to streamline this process by learning governing equations directly from timeseries. In mathematical epidemiology, such methods can fit data well but often produce differential equation models that violate conservation principles. We introduce a constrained sparse identification approach that preserves these quantities, yielding models that are both interpretable and faithful to epidemic dynamics.

In 1952, Alan Turing proposed that certain types of biochemical reactions, when coupled to molecular diffusion, can generate a stable spatial distribution of chemicals, such as stripes of high and low concentrations on a ring of cells or a dappled pattern on a sheet of cells. In an astounding anticipation of what we now call ‘molecular systems biology,’ Turing suggested that his theory might account for the ring-like tentacles of Hydra, the leaf whorls of Asperula, and the ‘dappled’ color patterns of animal coats. Turing’s theory of ‘morphogenesis’ was ignored for 15 years, until his ideas were revived by Prigogine and colleagues in Brussels and by Gierer and Meinhardt in Tübingen. Although Turing’s ideas are now standard fare in math biology courses, modern presentations of his theory are far removed from the 1952 paper because his linear stability analysis of the homogeneous steady state seems unnecessarily complex, his proposed examples are chemically unrealistic, and his numerical examples are perplexing. In this seminar, I propose to clear away some of these impediments and reintroduce Turing’s remarkable paper to a new generation of mathematical biologists.

The efficacy of Immune Checkpoint Inhibitors (ICIs) is frequently limited by adaptive immune resistance via the PD-1/PD-L1 pathway. While pairing ICIs with potent immunostimulants (such as NHS-muIL12) offers a strategy to overcome this, foundational mathematical models often struggle to capture complex preclinical behaviors, specifically non-monotonic tumor responses. In this talk, we introduce a refined deterministic model that addresses this limitation by incorporating PD-L1 expression as a fully dynamic variable rather than a static parameter. We demonstrate that this model, using a single parameter set, successfully reproduces experimental data across six distinct treatment arms, including monotherapies and combinations. This framework provides a robust mechanistic explanation for dose-dependent treatment failure driven by adaptive PD-L1 upregulation, while also elucidating the drivers of therapeutic synergy. We will present the systematic development of the model, simulation results, and a mathematical analysis of the system's stability and dynamics.

I introduce a set of phase reduction methods beyond weak perturbations for heterogeneous forced or coupled systems exhibiting n:m locking (where one oscillator traverses n periods in the same time that the other traverses m). The reduction enables a straightforward analysis of the existence and stability of n:m phase-locked states for general, high-dimensional oscillatory systems in much the same manner as classic weak coupling theory. I test the proposed method using high dimensional nonlinear, biologically relevant, (not necessarily) weakly attracting limit cycles (in particular from a circadian rhythm model and a thalamic neural oscillator model). I explore the accuracy and capabilities of the proposed reduction using various combinations of heterogeneous oscillators for varying coupling strengths and varying degrees of heterogeneity.

When do local interactions induce a global regime shift, and how can the underlying mechanism be made explicit? I introduce a Universal Dynamical Approximation (UDA) framework—a parameterized universality construction for finite-dimensional, structurally stable dynamical systems—that embeds diverse model classes into a common normal form. In these coordinates, the bifurcation structure provides a unified reduced description and yields reduced-order models for regime transitions. I show how UDA is realized in reaction–diffusion equations, interfacial flow models, center–satellite network ODEs, and stochastic lattice spin systems. Case studies span climate tipping points, ecosystem and evolutionary dynamics, and data-driven modeling, supported by modern data analysis and interpretable machine learning.

While immunotherapy has shown to be effective in treating some cancer types, the highly immunosuppressive tumor microenvironment of glioblastoma (GBM) provides unique challenges. As an example, immune checkpoint inhibitors, such as for the programmed-death-1 (PD-L1/PD-1) pathway, have had promising pre-clinical outcomes but failed to show efficacy in phase III clinical trials. One potential explanation is the infiltration of the tumor microenvironment by immune-suppressive cells such as myeloid-derived suppressor cells (MDSCs). Encouragingly, in experimental studies where MDSCs are targeted by drugs in combination with PD-1 blockade in mice, median survival increased. We develop an ODE model of the interactions between cancer cells, T cells, and MDSCs specific to the glioma microenvironment, and optimize combination immunotherapies for a virtual cohort of mice. We additionally develop an agent-based model to compare simulations to data extracted from cross-sectional tumor images of cellular biomarkers.

Self-organisation is a striking phenomenon that fascinates many. Especially in homogeneous environments, the question why - and how - heterogeneous structure emerges is intriguing. We find a novel function of self-organisation in biology at the colony margin of the fungus Aspergillus niger. Here, the collection of interconnected hyphae splits into two sub-populations with differential expression of amylolytic enzymes. These enzymes are secreted and together break down starch in a multi-step process. We show that concentrating all enzymes at part of the hyphae results in more efficient substrate degradation, as compared to equal distribution over all hyphae. The intermediate product maltose is known to induce upregulation of amylolytic enzymes. Incorporating this in the model enables the formation of hyphal heterogeneity through self-organisation.